The 3-sparsity of Xⁿ-1 over finite fields, II

2025/07/14 by Cheng, Kaimin
#11T06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.10779

Abstract

Let q be a prime power and \mathbbF_q the finite field with q elements. For a positive integer n, the polynomial Xⁿ - 1 ∈ \mathbbF_q[X] is termed 3-sparse over \mathbbF_q if all its irreducible factors in \mathbbF_q[X] are either binomials or trinomials. In 2021, Oliveira and Reis characterized all positive integers n for which Xⁿ - 1 is 3-sparse over \mathbbF_q when q = 2 and q = 4. Recently, the author provided a complete characterization for odd q. This paper extends the investigation to finite fields of even characteristic, fully determining all n such that Xⁿ - 1 is 3-sparse over \mathbbF_q for even q. This work resolves two open problems posed by Oliveira and Reis for even characteristic case.

Citations

• Closed formulas for the factorization of Xⁿ-1, the n-th cyclotomic polynomial, Xⁿ-a and f(Xⁿ) over a finite field for arbitrary positive integers n
• Factorization of composed polynomials and applications
• On explicit factors of Cyclotomic polynomials over finite fields

Related

• The 3-sparsity of Xⁿ-1 over finite fields
• Factors of Sparse Polynomials are Sparse
• Computing sparse multiples of polynomials
• Sparse Univariate Polynomials with Many Roots Over Finite Fields
• Further factorization of xⁿ-1 over a finite field